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% name as ATO: Morph Prime Knots 5 4 3.pdf

\title{\phantom{.} \vskip-3cm
\huge Morph Through Five Prime Knots \footnote{\large This file is from the 3D-XploreMath project. 
\hfil\break Please see http://www.math.uci.edu/$\sim$palais/  or http://3d-xplormath.org/}}
\author{}
\begin{document}

\maketitle
\vskip-2cm
\LARGE
A prime knot is a knot that cannot be written as the knot sum of smaller knots. For example,
the Square Knot and the Granny Knot are not prime since each is a sum of two Trefoil Knots. 
There are 14 prime knots with at most 7 minimal number of crossings. They have been hand
drawn so often that they have assumed an esthetically defined standard shape. Of these first
14 prime knots the following ones are in a morphing family, the prime knots 
$3_1, 4_1, 5_2, 6_1, 7_2$.  Choose $dd=3$ and $0 \le ff \le 4.3$ in Set Morphing and the 
program will deform the Trefoil Knot through the following images:
\vskip15pt

\hbox{\hskip-20pt
\vbox{\hsize=0.2\hsize \includegraphics[width=1.5 in]{MorphPrime3_1.png}}
\vbox{\hsize=0.2\hsize \includegraphics[width=1.5 in]{MorphPrime4_1.png}}
\vbox{\hsize=0.2\hsize \includegraphics[width=1.5 in]{MorphPrime5_2.png}}
\vbox{\hsize=0.2\hsize \includegraphics[width=1.5 in]{MorphPrime6_1.png}}
\vbox{\hsize=0.2\hsize \includegraphics[width=1.5 in]{MorphPrime7_2.png}}
}

If one chooses $dd=5$ and $0 \le ff \le 2.3$ in Set Morphing then the
program will deform the (5,2)-Torus Knot through the following images
of the prime knots $ 5_1, 6_2, 7_5 $:
\vskip15pt
\hbox{
\vbox{\hsize=0.2\hsize \includegraphics[width=1.5 in]{MorphPrime5_1.png}}
\vbox{\hsize=0.2\hsize \includegraphics[width=1.5 in]{MorphPrime6_2.png}}
\vbox{\hsize=0.2\hsize \includegraphics[width=1.5 in]{MorphPrime7_5.png}}
\hskip60pt
\vbox{\hsize=0.2\hsize \includegraphics[width=1.5 in]{LissajousKnot7_4.png}}
}
 
The prime knot $7_4$ is the default Lissajous space curve.
\goodbreak\eject
There are 249 prime knots with at most 10 minimal number of crossings. One can visualize
those via the Space Curves Menu entry: \break ``V. Jones Braid List''.
The notion of prime knot is important because Horst Schubert proved that the decomposition
of a knot as knot sum (= connected sum) of prime knots is unique. The knot invariants are
a good way to check whether a given knot is a prime knot.

There is an easy sufficient criterion that guarantees that the knot under consideration cannot
be drawn with fewer crossings. First we define {\it alternating} and {\it reduced alternating} knots:
if the thread of the knot passes alternatingly through
overcrossings and undercrossings then the knots is called {\bf alternating}.  For example, if
we twist a circle into a figure 8 we obtain an alternating trivial knot. In this case we observe an
easily recognizable property of the crossing in the knot diagram: if the crossing is removed the
knot diagram decomposes into two components. A crossing with this property is called an
{\it isthmus}. Clearly, one can always rotate one component of the knot diagram through 
180 degrees, i.e. untwist and thereby remove the isthmus to obtain a representation with fewer 
crossings. An alternating knot without an isthmus is called a {\it reduced alternating knot}.

{\it Theorem} :
Reduced alternating knots cannot be represented with fewer crossings, they are always non-trivial. 

All prime knots with at most 7 crossings are reduced alternating knots.

H.K.

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